Q. 24.1( 76 Votes )
In each of the following, determine whether the statement is true or false. If it is true, prove it. If it is false, give an example:
(i) If x ∈ A and A ∈ B, then x ∈ B
(ii) If A ⊂ B and B ∈ C, then A ∈ C
(iii) If A ⊂ B and B ⊂ C, then A ⊂ C
(iv) If A ⊄ B and B ⊄ C, then A ⊄ C
(v) If x ∈ A and A ⊄ B, then x ∈ B
(vi) If A ⊂ B and x ∉ B, then x ∉ A
Answer :
(i) Let us assume A = {1, 2}
And, B = {1, {1, 2}, {3}}
So, 
And, 
∴ 
But, 
Hence, the given statement is false
(ii) Let us assume, A {2}
B = {0, 2}
And, C = {1, {0, 2}, 3}
It is given in the question that,
∴ 
But, 
Hence, the given statement is false
(iii) It is given in the question that,
Let us assume, 
So, 
And, 
∴ 
Hence, the given statement is correct
(iv) It is given in the question that,
A ⊄ B
And, B ⊄ C
Let us now assume, A = {1, 2}
B = {0, 6, 8}
And, C = {0, 1, 2, 6, 9}
∴ 
Hence, the given statement is false
(v) It is given in the question that,
And, A ⊄ B
Let us now assume, A = {3, 5, 7}
And, B = {3, 4, 6}
As, 
And, 
∴ 
Hence, the given statement is false
(vi) It is given in the question that,
A ⊂ B
And, x ∉ B
Let us suppose, 
then we have:
But it is given in the question that, x ∉ B
∴ x ∉ A
Hence, the given statement is true
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